# NAME

Math::WalshTransform.pm - Fast Hadamard and Walsh Transforms

# SYNOPSIS

`````` use Math::WalshTransform;
@f = (1.618, 2.718, 3.142, 4.669);  # must be power-of-two long
@FH1 = &fht(@f);   # Hadamard transform
@fh1 = &fhtinv(@FH1);
# or
@FW2 = &fwt(@f);   # Walsh transform
@fw2 = &fwtinv(@FW2);

@PS  = &power_spectrum(@f);

import Math::WalshTransform qw(:ALL);
@whats_going_on = &biggest(9,&fwt(&sublist(\@time_series,-16)));
@EVENT1 = &fwt(&sublist(\@time_series,478,16));
@EVENT2 = &fwt(&sublist(\@time_series,2316,16));
@EVENT3 = &fwt(&sublist(\@time_series,3261,16));
\$EVENT1[\$[]=0.0; \$EVENT2[\$[]=0.0; \$EVENT3[\$[]=0.0; # ignore constant
@EVENT1 = &normalise(@EVENT1); # ignore scale
@EVENT2 = &normalise(@EVENT2);
@EVENT3 = &normalise(@EVENT3);
@TYPICAL_EVENT = &average(\@EVENT1, \@EVENT2, \@EVENT3);
...
@NOW = &fwt(&sublist(\@time_series,-16));
\$NOW[\$[] = 0.0;
@NOW = &normalise(@NOW);
if (&distance(\@NOW, \@TYPICAL_EVENT) < .28) { &get_worried(); }``````

# DESCRIPTION

These routines implement fast Hadamard and Walsh Transforms and their inverse transforms.

Also included are routines for converting a Hadamard to a Walsh transform and vice versa, for calculating the Logical Convolution of two lists, or the Logical Autocorrelation of a list, and for calculating the Walsh Power Spectrum - in short, almost everything you ever wanted to do with a Walsh Transform.

Intelligible speech can be reconstructed by transforming blocks of, say, 64 samples, deleting all but the several largest transform components, and inverse-transforming; in other words, these transforms extract from a time-series the most significant things that are going on. They should be useful for noticing important things, for example in software that monitors time-series data such as system or network administration data, share-price, currency, ecological, opinion poll, process management data, and so on.

As from version 1.10, Math::WalshTransform uses C routines to perform the transforms. This runs 25 to 30 times faster than previous versions.

Not yet included are multi-dimensional Hadamard and Walsh Transforms, conversion between Logical and Arithmetic Autocorrelation Functions, or conversion between the Walsh Power Spectrum and the Fourier Power Spectrum.

Version 1.17

# SUBROUTINES

Routines which take just one array as argument expect the array itself; those which take more than one array expect a list of references.

fht(@f);

The argument @f is the list of values to be transformed. The number of values must be a power of 2. fht returns a list @F of the Hadamard transform.

fhtinv(@F);

The argument @F is the list of values to be inverse-transformed. The number of values must be a power of 2. fhtinv returns a list @f of the inverse Hadamard transform.

fwt(@f);

The argument @f is the list of values to be transformed. The number of values must be a power of 2. fwt returns a list @F of the Walsh transform.

fwtinv(@F);

The argument @F is the list of values to be inverse-transformed. The number of values must be a power of 2. fwtinv returns a list @f of the inverse Walsh transform.

The argument @F is a Walsh transform; walsh2hadamard returns a list of the corresponding Hadamard transform.

The argument @F is a Hadamard transform; hadamard2walsh returns a list of the corresponding Walsh transform.

logical_convolution(\@x, \@y)

The arguments are references to two arrays of values x and y which must both be of the same size which must be a power of 2. logical_convolution returns a list of the logical (or dyadic) convolution of the two sets of values. See the MATHEMATICS section ...

logical_autocorrelation(@x)

The argument is a list of values x; the number of values must be a power of 2. logical_autocorrelation returns a list of the logical (or dyadic) autocorrelation of the set of values. See the MATHEMATICS section ...

power_spectrum(@x)

The argument is a list of values x; the number of values must be a power of 2. power_spectrum returns a list of the Walsh Power Spectrum of the set of values. See the MATHEMATICS section ...

# EXPORT_OK SUBROUTINES

The following routines are not exported by default, but are exported under the ALL tag, so if you need them you should:

`` import Math::WalshTransform qw(:ALL);``
biggest(\$k,@x)

The first argument \$k is the number of elements of the array @x which will be conserved; biggest returns an array in which the biggest \$k elements are intact and in place, and the other elements are set to zero. If \$k is 0 or negative, then biggest returns an array in which all elements less than the average (absolute) size have been set to zero.

sublist(\@array, \$offset, \$length)

This routine returns a part of the @array without, as splice does, munging the original array. It applies to arrays the same sort of syntax that substr applies to strings; the sublist is extracted starting at \$offset elements from the front of the array; if \$offset is negative the sublist starts that far from the end of the array instead; if \$length is omitted, everything to the end of the array is returned; if \$length is negative, the length is calculated to leave that many elements off the end of the array.

distance(\@array1, \@array2)

This routine returns the distance between the two arrays, according to the Euclidian Metric; in other words, the square root of the sum of the squares of the differences between the corresponding elements.

size(@array)

This routine returns the distance between the array and an array of zeroes, according to the Euclidian Metric; in other words, the square root of the sum of the squares of the elements.

normalise(@array)

This routine returns an array scaled so that its size is 1.0

average(\@array1, \@array2, ... \@arrayN)

This routine returns an array in which each element is the average of the corresponding elements of all the argument arrays.

product(\@array1, \@array2)

This routine returns an array in which each element is the product of the corresponding elements of the argument arrays.

# MATHEMATICS

The Hadamard matrix is a square array of plus and minus ones, whose rows and columns are orthogonal to each other. Hence, the product of the matrix and its tranpose is the identity matrix times a constant N which is equal to the order of the matrix. If N is a power of two, symmetrical Hadamard matrices can be defined recursively:

``````         | 1  1 |
2    | 1 -1 |

|    N     N |
|    N     N |``````

Another way to describe a Hadamard matrix of dimension 2^N x 2^N is that the entry in row i and column j is (-1)^P, where P is the number of positions in which the binary expansion of i and j share a 1. From this definition it is immediate that the last row (and column) is a Thue-Morse (or Morse-Thue) sequence, and also that rows that are of the form 2^N - 2^j will be j-fold repetitions of the Thue-Morse sequence.

The upper half of the Hadamard matrix are cycles of increasing wavelengths, and the lower half are Morse-Thue sequences on decreasing cell-sizes, much as the components of a Fourier analysis are sine-waves of different wavelengths.

The Walsh matrix is derived from the Hadamard matrix by rearranging the rows so that the number of sign-changes is in increasing order. Each row of the Walsh matrix corresponds to a Walsh Function Wal(j,k) where j = 0...N-1

The one-dimensional Hadamard transform pair is defined by

`````` F(j) = (1/N) * Sigma f(k)*Had(j,k)

and the one-dimensional Walsh transform pair is defined by

`````` F(j) = (1/N) * Sigma f(k)*Wal(j,k)
f(j) = Sigma F(k)*Wal(j,k)``````

The two transforms are equivalent, and conversion between them only involves rearranging the order of the components. Since the Walsh functions are in order of increasing number of sign-changes, the Walsh transform is more Fourier-like, and for that reason is used more often, although it does use several per-cent more CPU time.

Because all the matrix elements are either 1 or -1, these transforms involve almost no multiplications and are computationally very efficient.

The Logical (or Dyadic) Convolution of two arrays x and y is defined by

`` z(k) = (1/N) * Sigma x(k^j)*y(j)``

where the ^ is used in its Perl sense, to mean bitwise exclusive-or. There exists a Logical (or Dyadic) Convolution Theorem, analogous to the normal case, that the Walsh transform of the logical convolution of two sequences is the product of their Walsh transforms, and that the Walsh transform of the product of two sequences is the logical convolution of their Walsh transforms.

Likewise there exists a Logical Wiener-Khintchine Theorem, stating that the Walsh Power Spectrum is the Walsh transform of the Logical Autocorrelation Function.

There exist linear transformations converting between Logical Convolution and the normal Arithmetic Convolution, and between the Walsh Power Spectrum and the normal Fourier Power Spectrum.

# AUTHOR

Peter J Billam, www.pjb.com.au/comp/contact.html

Walsh Spectrometry, a form of spectral analysis well suited to binary computation, J. E. Gibbs, National Physical Lab, Teddington, Middlesex, England, unpublished, 1967

Hadamard transform image encoding, W. K. Pratt, J. Kane and H. C. Andrews, Proc. IEEE, Vol. 57, Jan 1969, pp. 58-68

Walsh function generation, D. A. Swick, IEEE Transactions on Information Theory (Corresp.), Vol. IT-15 part 1, Jan 1969, p. 167

Computation of the Hadamard transform and the R-transform in ordered form, L. J. Ulman, IEEE Trans. Comput. (Corresp.), Vol. C-19, Apr 1970, pp. 359-360

Computation of the Fast Hadamard Transform, Ying Shum and Ronald Elliot, Proc. Symp. Appl. Walsh Functions, Washington D.C., 1972, pp. 177-180

Logical Convolution and Discrete Walsh and Fourier Power Spectra, Guener Robinson, IEEE Transactions on Audio and Electroacoustics, Vol. AU-20 No. 4, October 1972, pp. 271-280

Speech processing with Walsh-Hadamard Transforms, Ying Shum, Ronald Elliot and Owen Brown, IEEE Transactions on Audio and Electroacoustics, Vol. AU-21 No. 3, June 1973, pp. 174-179

`````` http://www.pjb.com.au/